The French government has come under increasing pressure over an allegation that its budget minister, who is heading Paris's crackdown on tax evasion, once held a secret Swiss bank account hidden from the French authorities.

"I don't have, I never have had, accounts abroad," Cahuzac told the French parliament before the end of its December session when the story first ran. He dismissed Mediapart's assertion that he had travelled to Switzerland to close the Geneva-based account and transfer money to Singapore just before he was made head of the parliamentary finance commission in February 2010.

Cahuzac, a former MP and surgeon specialising in plastic surgery and hair transplants, has become one of the government's big-hitters as Paris struggles to redress its crisis-hit economy. After the Mediapart allegations, he received the backing of the prime minister, Jean-Marc Ayrault, who said he had "full confidence" in him. The Élysée has also stood by him.

But over the Christmas period the Cahuzac story has continued to dominate headlines as some newspapers suggested Hollande might have a cabinet reshuffle both to detract from the Mediapart allegations and to draw a line under government disagreements over the handling of France's crisis-hit steel industry. Several political analysts have since dismissed the idea of a reshuffle a mere seven months after Hollande took office.

The Socialist MP Olivier Faure told France 2 TV there was no concrete proof of the existence of a Swiss account of Cahuzac and that for now the "empty" allegations against the minister were "not very credible". He said if any irrefutable proof were ever produced then Cahuzac would be in difficulty and "would have to quit government", but for the moment nothing suggested the claims were true.

In 2010 Mediapart, which is run by veteran French newspaper and agency journalists, broke the story of an alleged party funding scandal around Nicolas Sarkozy's UMP party and the L'Oréal heiress Liliane Bettencourt. The website stood by its Cahuzac story, which it said was a question of "public morality" and the "exemplarity of a minister". It has called for an inquiry led by an independent judge.

The allegation against Cahuzac comes at a particularly sensitive time, as the government has increased income tax on the richest and criticised wealthy individuals, such as the actor Gérard Depardieu, who have sought to shift their tax base abroad.

Mediapart published a recording in which Cahuzac is alleged to be talking to his wealth manager about his embarrassment over the Swiss bank account in 2000. In the recording, a voice said to be Cahuzac's says: "What bothers me is that I still have an account open with UBS … UBS is not necessarily the most hidden of banks."

The conversation was said to have been taped inadvertently by Michel Gonelle, a barrister and one-time political opponent of Cahuzac in his south-western constituency. Gonelle has since called the Élysée to say the recording is authentic. Mediapart said Cahuzac had not issued legal proceedings over the recording, but the budget minister had told French radio journalists that it was not his voice.

Cahuzac remains in his government post and there is no sign of him moving.

The finance ministry denied another recent Mediapart report that its tax inspectors were investigating Cahuzac's French assets. In a statement, the ministry said the public finances directorate was examining the tax situation of all government members as a matter of "routine", with the aim of "ensuring that the position of every government member is irreproachable".

The finance ministry added: "No inspection or investigation is in progress with regard to any government member."